One begins by taking the logarithm of both sides of equation (1), yielding
\log z = z\log i. \tag2
(Incidentally, this is reminiscent of Lambert's w-function.)
As \log i is multi-valued, this becomes
\frac{1}{z} \log z = (4n +1)\frac{\pi}{2} i. \tag3
Setting z = re^{i\theta}, expanding with Euler's formula for e^{i\theta},
and equating the real and imaginary parts, one finds
\frac{1}{r}(\cos\theta.\log r + (\theta + 2k\pi).\sin\theta) = 0, \tag4
\frac{1}{r}\left( (\theta + 2k\pi).\cos\theta - \sin\theta.\log r \right) =
(4n+1)\frac{\pi}{2}.
\tag5
Equation (4) furnishes a formulae for r in terms of \theta:
\log r = - (\theta + 2k\pi)\tan\theta. \tag6
Using this once only in equation (5) furnishes a more instructive formula for
r
in terms of \theta:
r = \frac{2(\theta + 2k\pi)}{(4n+1)\pi\cos\theta}\;, \tag7
while repeated substitution of equation (6) into equation (5) gives an implicit
formula for \theta:
\frac{(\theta + 2k\pi)e^{(\theta + 2k\pi)\tan\theta}}{\cos\theta} =
(4n+1)\frac{\pi}{2}.
\tag8
Clearly for chosen values of k and n, equation (8) can be solved numerically
for \theta. The value found can then be used in equation (7), or (6), to find
r.
Our particular interest in evaluating i^\wedge i^\wedge i^\wedge i^\wedge
\ldots was to search for patterns in the (infinite number of) solutions. For
example with k=0, solutions to equation (8) are graphed as shown in figure~ 1.
The
multiplicity of solutions is evident, as is the asymptotic behaviour of these
solutions:
\theta \sim 2m\pi. \tag9
Because of this, \cos\theta \sim 1, whereupon equation (7) gives
r \sim \frac{2}{(4n+1)\pi}\theta. \tag10
That is, on a (r,\theta) diagram, the solutions are asymptotically on a
straight line. For other values of k, the solutions are asymptotically upon the
straight lines
r = \frac{2}{(4n+1)\pi}(\theta+ 2k\pi). \tag11
To next order, one can solve equation (8) to find
\theta^{(2)} \sim 2m\pi - \frac{1}{2(m + k)\pi} \log\left(\frac{4(m + k)}{4n +
1}\right)
\tag12
and equation (11) still holds, but with \theta^{(2)} inserted in place of
\theta. Clearly this could be continued to any order.
Complementary relationships emerge in the cartesian plane. Setting z = x + iy
and substituting (7) into the transformation:
x = r\cos\theta, \qquad\qquad y = r\sin\theta,
\tag13
gives:
x = \frac{2(\theta + 2k\pi)}{(4n + 1)\pi} \qquad \text{and} \qquad y =
\frac{2(\theta + 2k\pi)}{(4n+1)\pi}\tan\theta.
\tag14
Expanding (14) for \theta close to 2m\pi and evaluating one finds:
y \sim \frac{-2}{(4n + 1)\pi} \log x.
\tag15
We note that (15) is valid for any k.
This note has shown that there are infinitely many values of i^\wedge i^\wedge
i^\wedge i^\wedge \ldots and that these values are distributed in the complex
plane in a reasonably simple way.
Department of Mathematics
University College, UNSW
ADFA Canberra 2600